8. Proof techniques and examples

Problem 8.1.
Is is possible to show that \begin{equation*} \int_X^{2X} \left \sum_{x\leq n\leq x+H} \lambda(n) e(n \alpha) \right d ̣x = o(XH) \end{equation*} with $H > X^\epsilon$ implies \begin{equation*} \sum_{n \leq X} \frac{\lambda(n)\lambda(n+1)}{n} = o(\log X)? \end{equation*} 
Problem 8.2.
Does the Sarnak conjecture hold uniformly in $x \in X$ in the case of horocycle flows? It is known that the uniform Sarnak conjecture follows from the Sarnak conjecture, but we don’t know that it follows case by case. 
Problem 8.3.
The Chowla conjecture implies that $\mu$ cannot be orthogonal to all uniquely ergodic systems (allowing nonzero entropy). Can we find a concrete example of a uniquely ergodic system that correlates with $\mu$? A way to find a uniquely ergodic system (of positive entropy) that does not correlate with $\lambda$ is known.
Remark. [Nikos] Since the system of Problem 3.1 is disjoint from all ergodic systems, an example of an ergodic sequence (or uniquely ergodic system) that correlates with the Liouville function would give a negative answer to Problem 3.1.

Cite this as: AimPL: Sarnak's conjecture, available at http://aimpl.org/sarnakconjecture.